Problem: Solve the equation. $\dfrac{dy}{dx}=-\dfrac{e^x y^2}{10}$ Choose 1 answer: Choose 1 answer: (Choice A) A $y=\dfrac{10}{e^x}+C$ (Choice B) B $y=\dfrac{10}{e^x+C}$ (Choice C) C $y=\dfrac{10}{Ce^x}$ (Choice D) D $y=\dfrac{10+C}{e^x}$
Solution: We can bring this equation to the form $f(y)\,dy=g(x)\,dx$ : $\begin{aligned} \dfrac{dy}{dx}&=-\dfrac{e^x y^2}{10} \\\\ -\dfrac{10}{y^2}\,dy&=e^{x}\,dx \end{aligned}$ This means we can solve this equation using separation of variables! $\begin{aligned} -\dfrac{10}{y^2}\,dy&=e^{x}\,dx \\\\ \int -\dfrac{10}{y^2}\,dy&=\int e^{x}\,dx \\\\ \dfrac{10}{y}&=e^x+C \\\\ \dfrac{1}{y}&=\dfrac{e^x+C}{10} \\\\ y&=\dfrac{10}{e^x+C} \end{aligned}$ Notice that after the integration, more work was required in order to isolate $y$. In conclusion, this is the solution of the equation: $y=\dfrac{10}{e^x+C}$